Gröbner Bases, Ideal Computation and Computational Invariant Theory

Transcription

1 Gröbner Bases, Ideal Computation and Computational Invariant Theory Supervised by Kazuhiro Yokoyama Tristan Vaccon September 10,

2 Contents Preamble 4 Introduction 4 1 Algebraic Geometry Affine Varieties Definition Ideals and Affine Varieties Hilbert s Nullstellensatz Preliminary work Weak form Strong form Links between Algebra and Geometry Zariski Topology Operations on Ideals and Varieties Gröbner Bases Preliminary Works Monomial Ordering Division Algorithm Monomial Ideals and Dickson s lemma Definition and Properties Definition Division and Gröbner Bases How to Compute a Gröbner Bases Buchberger s Algorithm Minimal and Reduced Gröbner Bases Ideal Computation with Gröbner Bases Elimination Ideal Intersection of Ideals A formula How to add a new variable, coercion in Magma An Algorithm in Magma Ideal Quotient and Saturation Ideal Quotient Saturation The Dimension of an Ideal Definitions and first properties

3 3.4.2 Heuristics An algorithm The Radical of an Ideal Zero-dimensional Radical in Zero Characteristic Higher-dimensional Radical in Zero Characteristic Radical in positive characteristic Computation Results and Examples Basic operations About Dimension Computation About Radical Computation Computational Invariant Theory Symmetric Polynomials Some definitions The fundamental theorem of symmetric polynomials Miscellaneous Ring of Invariants under the action of finite matrix groups Some definitions Generators of the ring of invariants Hilbert Series and Molien s Formula Hilbert Series Molien s Formula The Hilbert Series of a Finitely Graded Algebra Primary and Secondary Invariants Some Definitions An Algorithm to Compute Primary Invariants An Algorithm to Compute Secondary Invariants Computational Results and Exemples An exemple of computation of the invariant ring under the action of a not-necessarily finite group Conclusion 52 Thanks 52 References 53 Annex : Implementation in Magma 54 3

4 Preamble My internship took place at Rikkyo University (Ikebukuro, Tokyo, Japan) between the first of June and the 31st of July It was supervised by Prof. Kazuhiro Yokoyama. I had my desk at the graduate students s room and twice a week, I had to give a one-hour presentation in front of Prof. Yokoyama and sometime some of his students and post-docs. My path to Computational Invariant Theory during the internship was really close to the one given in this report. Introduction We will indeed first recall some basic knowledge about algebraic geometry, and then, some basic knowledge about Gröbner bases. The main highlight of those two first parts should be the very elementary proof of Hilberts Nullstellensatz we will give. After that, we will study Computational Ideal Theory, id est how operations on ideals can be computed (with the help of Gröbner bases), and finally, Computational Invariant Theory. We will also consider implementation of the algorithms we will study. All the implementation I made were in Magma, and they all can be found in the Annex. Remark. Unlike what might be used outside France, in what follows, N will denote the set of natural numbers, and N the set of positive natural numbers. 1 Algebraic Geometry In this first section, our main goal would be to prove Hilbert s Nullstellensatz, and present the links between algebra and geometry. Since it is very basic knowledge on algebraic geometry, many proofs (either obvious or very classical) will not be given. Yet, most of them can be found in Cox, Little & O Shea [1]. 1.1 Affine Varieties Definition Definition 1. Let k be a field and let F k [x 1,..., x n ] (n N ). Then we call V (F ) = {(a 1,..., a n ) k n f F, F (a 1,..., a n ) = 0} the affine variety defined by F. Remark. We have V (F ) = V (I(F )) where I(F ) is the ideal spanned by F. 4

5 1.1.2 Ideals and Affine Varieties Definition 2. Let I be an ideal of k [x 1,..., x n ], then {f 1,..., f s } I is called a basis of I if f 1,..., f s = I. Definition 3. Let A be a commutative ring. A is called noetherian if it satisfies the ascending chain condition : given any chain of ideals of A, I 1 I k I k+1..., then there exists n N such that I n = I n+1 =.... It is equivalent to the fact that every non-empty set of ideals of A has a maximal element. Proposition 1. A is noetherian if and only if all ideals of A are finitely generated. Theorem 1 (Hilbert s Theorem). If A is noetherian, then A[X] is noetherian. Thus, n N, A [X 1,..., X n ] is noetherian. Remark. A field is noetherian (it contains only two ideals...), thus if k is a field, then n N, k [X 1,..., X n ] is noetherian. Definition 4. Let n N and V k n. We set I(V ) = {f k [X 1,..., X n ] (a 1,..., a n ) V, f(a 1,..., a n ) = 0} Lemma 1. I(V ) is an ideal of k [X 1,..., X n ], and is called the ideal generated by V. Lemma 2. If I J are ideals of k [X 1,..., X n ] then V (I) V (J). Lemma 3. If V W k n, then I(V ) I(W ). Lemma 4. If J is an ideal of k [X 1,..., X n ] then J I(V (J)). Now that we have recalled those basic definitions, we shall give a proof of Hilbert s Nullstellensatz, which might be the shortest basic proof (id est that does not need some more advanced mathematics) currently known that stands for any algebraically closed field. 1.2 Hilbert s Nullstellensatz Preliminary work Definition 5. Let A be a ring. B is a finitely generated A-algebra if there exists n N and an ideal I of A [X 1,..., X n ] so as B A [X 1,..., X n ] /I. If so, B is called integral if any element of B is integral over A, id est for all x B, there exists a monic f A[X] with f(x) = 0. 5

6 Lemma 5. Let A and B be two rings with B integral over A. Then if B is a field, A is also a field. Proof. Let x A\{0}. y = x 1 B is integral over A, so there exists a monic f A[X] with f(y) = 0, id est there exists n N and a 0,..., a n 1 with y n +a n 1 y n 1 + +a 0 = 0. Multiplying by x n 1 we obtain y = (a n a 0 x n 1 ), and thus, y A. Proposition 2. Let k be a field, and B a k-algebra finitely generated. If B is a field, then B is a finite algebraic extension of k. Proof. We proceed by induction. For n N, let P (n) be the proposition "If k is a field and B a k-algebra generated by n elements, and which is a field, then B is a finite algebraic extension of k." P (0) is true, since in that case B = k. If n = 1, then we have B = k[x 1 ] for some x 1 in B. If x 1 is transcendental over A, then B k[x], which is not a field. So, x 1 is algebraic over A, and then, B is a finite algebraic extension of k. Now, we assume that n > 1 and P (n 1) is true. Let B be a k-algebra generated by n elements, so B = k [x 1,..., x n ] for some x 1,..., x n in B. Let A = k[x 1 ] and let K be the fraction ring of A. Since B is a field and A B, then K B. Thus, B is a K-algebra generated by n 1 elements. With P (n 1), B is a finite algebraic extension of K. So x 2,..., x n are algebraic over K. Hence, there exist P i K[X], monics, so that P i (x i ) = 0, 2 i n. Let f k[x 1 ] be the product of the denominators of the coefficients of the P i. Then x 2,..., x n are integral over A f = A[1/f], and since its generator are integral over A f, so is B. Since B is a field, A f B. With the previous lemma, A f is a field. If x 1 is transcendental over k, then A f = k[x][1/p ] for some P k[x], nonzero. Yet, 1 XP is a non-zero polynomial in k[x] and thus in A f, but it has no inverse in A f : if there exists Q A f so as Q(1 XP ) = 1, then the evaluation in X = 1/P leads to an absurdity. So, 1 XP has no inverse in A f while A f is a field, which is absurd. Thus, x 1 is algebraic over k. It follows that K is is a finite algebraic extension of k, and since B is a finite algebraic extension of K, we can deduce the result Weak form Theorem 2. Let n N and I be a maximal ideal of k [X 1,..., X n ], where k is an algebraically closed field. Then there exists (a 1,..., a n ) k n such that I = X 1 a 1,..., X n a n. Proof. First, we shall prove that the ideals of the form J = X 1 a 1,..., X n a n, with (a 1,..., a n ) k n, are maximals. Obviously, φ : k [x 1,..., x n ] k, defined 6

7 by φ(p ) = P (a 1,..., a n ) is a surjective ring homomorphism. If P k [X 1,..., X n ] and if we apply the euclidean division of P by X 1 a 1 in k [X 2,..., X n ] [X 1 ], and then the euclidean division of the remainder (which lies in k [X 2,..., X n ]) by x 2 a 2, in k [X 3,..., X n ] [X 2 ],..., we obtain P = (X a 1 )Q (X n a n )Q n + P (a 1,..., a n ), with Q j k [X j,..., X n ]. So if P Kerφ, then P J, and Kerφ J. Of course, J Kerφ, and Kerφ = J, k [X 1,..., X n ] /J = k and J is a maximal ideal. Now, let M be a maximal ideal of k [X 1,..., X n ], then k [X 1,..., X n ] /M is a finitely generated k-algebra which is a field. With the previous proposition, it is an algebraic finite extension of k, and since k is algebraically closed, k [X 1,..., X n ] /M = k. So for i {1,..., n}, there exists a i k such that X i a i M. Then X 1 a 1,..., X n a n M, and we have already seen that X 1 a 1,..., X n a n is maximal. Hence the result is proven. Theorem 3 (The Weak Nullstellensatz). Let k be an algebraically closed field and let I k [X 1,..., X n ] be an ideal so as V (I) =. Then I = k [X 1,..., X n ]. Proof. With the fact that k [X 1,..., X n ] is noetherian, if I k [X 1,..., X n ] then there exist a maximal ideal M of k [X 1,..., X n ] such that I M. With the previous proposition, there exists (a 1,..., a n ) k n such that X 1 a 1,..., X n a n M. Yet, I M so (a 1,..., a n ) V (M) V (I) and V (I) which is absurd. So I = k [X 1,..., X n ]. Remark. It is a little bit different but we can also show that if k is an infinite field (which is the case for an algebraically closed field), and if P k [X 1,..., X n ], with P 0, then there exists x 1,..., x n such that P (x 1,..., x n ) 0. Proof. Let us show by induction on n that if P k [X 1,..., X n ], with x k n, P (x) = 0, then P = 0. When n = 1, the result is well-known. We assume that n 2 and that the result is proven for n 1. Let P k [X 1,..., X n ], with x k n, P (x) = 0. We can write P = N i=1 A i X i n, with A i k [X 1,..., X n 1 ]. If (x 1,..., x n 1 ) k n 1, then we have for all x n k, P (x 1,..., x n ) = 0 = N i=1 A i (x 1,..., x n 1 ) x i n. So with the case n = 1, the polynomial in X n, N i=1 A i (x 1,..., x n 1 ) X i n, equals the zero polynomial. Hence, If (x 1,..., x n 1 ) k n 1, then A i (x 1,..., x n 1 ) = 0. With the case n 1, A i = 0, and thus, P = 0. Finally, the result is proven Strong form Definition 6. Let I be an ideal of k [X 1,..., X n ]. We define its radical ideal I by I = {f k [X1,..., X n ] m N, f m I}. 7

8 An ideal I such that I = I is called radical. Theorem 4 (Hilbert s Nullstellensatz). Let k be an algebraically closed field. Let J be an ideal of k [X 1,..., X n ]. Then J = I(V (J)). Proof. First, let P J, then there exist r N such that P r J. Then, if x V (J), then P r (x) = 0 and k is a field so P (x) = 0. Thus, P I(V (J)) and J I(V (J)). Conversely, let P I(V (J). In k [X 1,..., X n, T ], let J be the ideal generated by J and 1 T P. If V (J ), let z = (x 1,..., x n, t) = (x, t) V (J ). Then (1 P T )(z) = 0 and also x V (J), so P (x) = 0. Thus (1 P T )(z) = 1 P (x)t = 1 0 which is absurd. So V (J ) =. With the previous theorem, J = k [X 1,..., X n, T ], so there exist m N, Q 0,..., Q m in k [X 1,..., X n, T ], (P 1,..., P j ) J m such that m (1 T P )Q 0 + Q j P j = 1. j=1 In k (X 1,..., X n ), we can substitute T by 1 P and we find m Q j (X 1,..., X n, 1 j=1 P )P j(x 1,..., X n ) = 1. With r large enough (for instance, larger than the maximum over j of the maximum of the degree in T of the Q j ), we have the equality in k [X 1,..., X n ] : P r = m j=1 P r Q j (X 1,..., X n, 1 P )P j(x 1,..., X n ). Hence, P r J, and the result is proven. With this theorem, it is easy to deduce a first correspondence between geometry (varieties) and algebra (ideals) : Definition 7. A topological space is called irreducible if it cannot be expressed as the union of two proper closed subset. A nonempty subset of a topological space is called irreducible if it is an irreducible topological space for the induced topology. The empty set is not considered irreducible. Corollary 1. There is a one-to-one inclusion-reversing correspondence between affine varieties of k n and radical ideals, given by V I(V ) and I V (I). There is also a one-to-one correspondence between irreducible affine varieties and prime ideals. Remark. k n is irreducible for the Zariski topology since 0 is prime (k [X 1,..., X n ] being indeed an integral domain). 8

9 1.3 Links between Algebra and Geometry Zariski Topology Let n N. We can define a topology on k n by taking the closed sets as the affine varieties (defined with ideals of k [X 1,..., X n ]), which is called the Zariski topology on k n. Proposition 3. The Zariski topology on k n is well defined. Proof. = V (k [X 1,..., X n ]) ; k n = V (0) ; let I and J be two ideals of k [X 1,..., X n ]. Then V (I) V (J) = V (IJ), and every finite union of affine varieties is an affine variety ; if (I α ) α I are some ideals of k [X 1,..., X n ], then α I V (I α ) = V (+ α I I α ). Thus, the Zariski topology is a a topology on k n. Now, we can state what happens with the applications V I(V ) and I V (I) when we do not deal with necessarily affine varieties : Lemma 6. If A k n, then V (I(A)) = A. Proof. Let f I(A) and a A. Then f(a) = 0 and thus a V (I(A)), and V (I(A)) A. Since V (I(A)) is closed, V (I(A)) A. By definition, there exists some ideal J such that A = V (J). Then V (I(A)) V (J) A, thus (I(A)) (J) A. Hence, V ( J) V (I(A)) and V ( J) = V (J) = A. Finally, V (I(A)) = A Operations on Ideals and Varieties Definition 8. If I and J are ideals of k [X 1,..., X n ], then we define I : J with I : J = {f k [X 1,..., X n ] g J, fg I}, and I : J is called the ideal quotient, or the colon ideal, of I by J. Remark. If I and J are ideals of k [X 1,..., X n ], then I : J is indeed an ideal of k [X 1,..., X n ]. Proposition 4. If k is algebraically closed and I and J are ideals in k [X 1,..., X n ], then : 9

10 V (I + J) = V (I) V (J) ; V (IJ) = V (I J) = V (I) V (J), with IJ = fg, (f, g) I J ; I J = I J ; V (I : J) V (I) V (J) (with A B = A B c ), and if in addition, I is a radical ideal, then V (I : J) = V (I) V (J). Proof. We wil only prove the last property. First, we show that I : J I(V (I) V (J)). If V (I) V (J), the result is obvious. Let f I : J and x V (I) V (J), then fg I for all g J. Since x V (I), we have f(x)g(x) = 0 for all g J. x / V (J) so there exists g V (J) such that g(x) 0. Hence, with f(x)g(x) = 0, f(x) = 0, and then I : J I(V (I) V (J)). Hence, V (I : J) V (I(V (I) V (J))) = V (I) V (J). Then, with I a radical ideal, let x V (I : J). Let h I(V (I) V (J)). If g J, then g vanishes on V (J), h on V (I) V (J) so hg vanishes on V (I). Hence hg I = I. So, if h I(V (I) V (J)), if g J, hg I, and thus h I : J. Hence, I(V (I) V (J)) I : J. Thus, I : J = I(V (I) V (J)), and then V (I : J) = V (I) V (J). Now that the theoretical basis needed are recalled, we can wonder how computations can be made with ideals and such operations (radical, intersection, quotient,...), and hence come Gröbner bases. 2 Gröbner Bases Before being able to define what a Gröbner Basis is, we shall present some preliminary results and definitions. Again in this section, some proofs will not be given, and we refer for them to Cox, Little & O Shea [1]. 2.1 Preliminary Works Monomial Ordering Definition 9. A monomial ordering on N n (n N ) is an ordering > satisfying : > is a total ordering on N n ; if α > β and γ N n, then α + γ > β + γ ; > is a well-ordering on N n, which means that every nonempty subset of N n has a smallest element for >. 10

11 Remark. > is a well-ordering on N n if and only if there is no strictly decreasing sequence (for >) in N n. Remark. A monomial on N n order clearly induce a total order on the monomials of k [X 1,..., X n ]. Definition 10 (Lexicographical order). We define the lexicographical order > lex on N n with : if (α, β) N n N n, then α > lex β if the left-most nonzero coordinate of α β is positive. Definition 11 (Graded Lex Order). We define the graded lexicographical order > grlex on N n with : if (α, β) N n N n, then α > grlex β if α > β, or α = β and α > lex β. Definition 12 (Graded reverse Lex Order). We define the graded reverse lexicographical order > grevlex on N n with : if (α, β) N n N n, then α > grevlex β if α > β, or α = β and the right-most nonzero coordinate of α β is negative. Proposition 5. All of the three orders above are monomial orders. Hence, we can define what is the multidegree of a polynomial, its leading coefficient, monomial,... Definition 13. Let f = α a α x α be a nonzero polynomial in k [x 1,..., x n ], and let > be a monomial order. Then : The multidegree of f is multidegree(f) = max (α N n a α 0) (maximum taken with respect to >). The leading coefficient of f is LC(f) = a multideg(f) k. The leading monomial of f is LM(f) = x multideg(f). The leading term of f is LT (f) = LC(f) LM(f) Division Algorithm We can generalize the euclidean division algorithm of k[x] to k [x 1,..., x n ]. Theorem 5. Being given a monomial order > on N n and F = (f 1,..., f s ) be an ordered s-tuple of polynomials in k [X 1,..., X n ], if f k [X 1,..., X n ], then f can be written f = a 1 f a s f s +r where all a i and r are in k [X 1,..., X n ], and either r = 0 or none of its monomials is divisible by any of the LT (f 1 ),..., LT (f s ). r is called a remainder of f on division by F (there is no unicity), and if a i f i 0 then multideg(f) multideg(a i f i ) for all i. The a i can be computed by the algorithm given below. 11

12 The idea of this algorithm is really the same than the one for k[x] : trying to eliminate the leading term (with respect to the order given) possible of f with the leading terms of F (F being ordered), and if it is not possible, this leading term goes to the remainder. Algorithm 1 Division algorithm in k [X 1,..., X n ] of f by (f 1,..., f s ) r:=0; p:=f; for i in [1..s] do a i := 0; end for while p 0 do i:=1; divisionoccured:=false; while i s & divisionoccured == false do if LT (f i ) divides LT (p) then LT (p) a i := a i + LT (f i ) ; LT (p) p := p f LT (f i ) i; divisionoccured:=true; else i:=i+1 end if end while if divisionoccured==false then r:=r+lt(p); p:=p-lt(p); end if end while return a 1,..., a s, r Monomial Ideals and Dickson s lemma Since in the division algorithm, it is mainly leading terms from F that are operating, we should consider the so-called monomials ideals : Definition 14. An ideal I k [x 1,..., x n ] is a monomial ideal if it can be generated by a set of monomials : thus if I = x α, α A for some A N n. Lemma 7. Let I = x α, α A be a monomial ideal. Then x β, for some β N n, lies in I if and only if x β is divisible by x α, for some α A. 12

13 Lemma 8. Let I be a monomial ideal, and f k [x 1,..., x n ], then f I if and only if every term of f lies in I, if and only f is a k-linear combination of the monomials in I. Lemma 9. Two monomials ideals are the same if and only if they contains the same monomials. Then, the main result for monomial ideals is Dickson s lemma : Theorem 6 (Dickson s Lemma). A monomial ideal I = x α, α A k [x 1,..., x n ] can be written in the form I = x α 1,..., x αs, for some α i A. Proof. We proceed by induction on the number of variables, n. If n = 1, then I is genererated by the x α, α A N. If β is the smallest element of A, then x β divides all the x α with α A, and I = x β Now, we assume that the theorem is true for n 1, for some n N, n 2. Suppose that I is a monomial ideal of k [x 1,..., x n, y]. We will write the monomials of k [x 1,..., x n, y] as x α y m with α N n 1 and m N. Let J be the ideals in k [x 1,..., x n ] defined by the monomials x α for which there exists m N such that x α y m I. J is a monomial ideal and by the inductive hypothesis, we can write J = x α(1),..., x α(s). For i {1,..., s}, there exists m i N such that x α(i) y m i I. Let m be the maximum of the m i. Now, for each k {0,..., m 1}, we consider J k the ideal in k [x 1,..., x n ] defined by the monomials x β such that x β y k I. Again, J k = x α k(1),..., x α k(s k ). Finally, let us show that I is generated by the x α(1) y m,..., x α(s) y m (coming from J), and all the x α k(1) y k,..., x α k(s k ) y k coming from J k, for each k {0,..., m 1}. Of course, by definition, all those monomials are in I. If x α y p is a monomial of I, then : if p m, then by definition of J, x α y p is divisible by some x α(i) y m. elsewhere, p m 1, then x α y p is divisible by some x αp(i) y p. Hence, those monomials generate an ideals having the same monomials as I, and thus by a previous lemma, I is generated by those monomials. Finally, we shall prove that those generators can be chosen in A. If I = x α, α A, we have seen that we can write I = x β 1,..., x βp. For i {1,..., p}, since x β i I, x β i can be divided by some x α i, α i A. Hence, I = x α 1,..., x αp. But what is more impressive is that this property somehow still holds for any ideal, and here will lie the idea of Gröbner bases. 13

14 2.2 Definition and Properties Definition Definition 15. Let I k [x 1,..., x n ] be an ideal, different from {0}. Given a monomial order, we define LT (I) = {LT (f) f I. We will naturally note LT (I) the ideal generated by LT (I). Lemma 10. LT (I) is a monomial ideal, and thus, there exists g 1,..., g t in I such that LT (I) = LT (g 1 ),..., LT (g t ). Definition 16. Given a monomial order, a finite subset G = {g 1,..., g t } of an ideal I is called a Gröbner Basis if LT (g 1 ),..., LT (g t ) = LT (I). Proposition 6. Given a monomial order, every ideal I k [x 1,..., x n ] has a Gröbner basis, and any Gröbner basis of I generates I. Proof. The previous lemma show the first part of the proposition. For the second part, if g 1,..., g t is a Gröbner basis of an ideal I, then, if f I, the division of f by g 1,..., g t, with the division algorithm we have seen previously, gives us f = a 1 g a t g t + r, with no term of r divisible by one of the LT (g i ). Hence, r = 0 because by construction, r I, so LT (r) LT (I) = LT (g 1 ),..., LT (g t ) and thus, r should be divisible by some LT (g i ), which is absurd. Finally, f g 1,..., g t, and the result is proven Division and Gröbner Bases Proposition 7. Let G = {g 1,..., g t } be a Gröbner basis of an ideal I k [x 1,..., x n ], and let f k [x 1,..., x n ], then there is a unique r k [x 1,..., x n ] such that no term of r is divisible by any of the LT (g 1 ),..., LT (g t ), and there is g I with f = g + r. In particular, r is the remainder of the division of f by G, no matter of how the elements of G are listed for the division algorithm, and f I if and only if the remainder of the division of f by G is 0. Proof. With the division algorithm, the existence of such a f is provided. Considering uniqueness, if f = g (1) + r 1 = g (2) + r 2 with r 1 r 2, then r 1 r 2 = g (2) g (1) I, and thus, LT (r 1 r 2 ) is divisible by some LT (g i ), which is impossible since no term of r 1 nor r 2 is divisible by any of the g 1,..., g t. Thus, r 2 = r 1 and the uniqueness is proven. Definition 17. Let f, g k [x 1,..., x n ] be two nonzero polynomials. 14

15 If multideg(f) = α and multideg(g) = β, let γ = (γ 1,..., γ n ), where γ i = max(α i, β i ) for each i. Then x γ is called the least common multiple of LM(f) and LM(g) : x γ = LCM(LM(f), LM(g)). The S-polynomial (from syzygy polynomial) of f and g is S(f, g) = xγ LT (f) f xγ LT (g) g Theorem 7. Let I be an ideal of k [x 1,..., x n ], then a basis G = {g 1,..., g t } for I is a Gröbner basis for I if and only if for all i, j in {1,..., t}, the remainder of the division of S(g i, g j ) by G is zero. 2.3 How to Compute a Gröbner Bases Buchberger s Algorithm The previous theorem gives us an algorithm to compute a Gröbner basis of an ideal, given a basis of that ideal : this is the Buchberger s algorithm. Algorithm 2 Buchberger s algorithm : computes a Gröbner basis of the ideal I = F, F = (f 1,..., f s ) G := F ; repeat G := G ; for (p, q) G 2, p q do S := S(p, q) G if S 0 then G := G {S} end if end for until G == G return G Minimal and Reduced Gröbner Bases The results given by Buchberger s algorithm can be very big in size, and some of its elements can be somehow "useless". What follow will explain this idea. Lemma 11. Let G be a Gröbner basis for the polynomial ideal I, and let p G be a polynomial such that LT (p) LT (G \ {p}). Then G \ {p} is also a Gröbner basis for I. 15

16 Definition 18. A minimal Gröbner basis for a polynomial ideal I is a Gröbner basis G for I such that : p G, LC(p) = 1 p G, LT (p) / LT (G \ {p}) Definition 19. A reduced Gröbner basis for a polynomial ideal I is a Gröbner basis G for I such that : p G, LC(p) = 1 For all p G, no monomials of p lies in LT (G \ {p}) Proposition 8. Let I {0} be a polynomial ideal. Then, for a given monomial ordering, I has a unique reduced Gröbner basis. Yet, we have chosen not to look for an algorithm that gives minimal or reduced, and instead, we have prefered jumping directly to the applications of Gröbner basis to the ideal computation, and then, computational invariant theory. This choice will reveal mostly harmless. Still, some of the more advanced algorithm might suffer a little from the use of non-reduced and non-minimal Gröbner basis in their implementation. 3 Ideal Computation with Gröbner Bases Gröbner basis are the tool "par excellence" of any computation involving ideals. In this section, we will particularly see how Gröbner basis can be used to compute operations on ideals : intersection, ideal quotient, Elimination Ideal The first operation that we will need to compute is that of elimination ideal, and to begin with, we shall of course define what an elimination ideal is : Definition 20. Let I k[x 1,..., X n ] be an ideal, and U {X 1,..., X n }. Then the ideal I k[u] in k[u] is called an elimination ideal. If U = {X i,..., X n }, then the ideal I k[u] in k[u] is called the i-th elimination ideal. Elimination ideals, and the fact that they can be very easily computed, will prove very useful for more complicated and avanced computation. The proposition that follows explain how they can indeed be computed. 16

17 Proposition 9. Let I k[x 1,..., X n ] be an ideal, and U {X 1,..., X n }. Let G be a Gröbner basis of I with respect to a monomial order such that for all s k[u], t k[u c ] non constant, u k[u], s < ut. Then G K[U] is a Gröbner basis of the elimination ideal I k[u], with respect to the order induced on k[u]. Proof. The first thing that we can notice is that with such a monomial ordering, and with G a Gröbner basis with respect to that order, if we take P G k[u] c, then necessarily, its leading monomial would have nonzero exponents in U c since if no term of P has nonzero exponents in U c, then P k[u], and with the fact that s k[u], t k[u c ], u k[u], t non constant, s < ut, then any such term is greater than any term in k[u], and thus the leading monomial of P can not lie in k[u]. Then, if x I k[u], and if we perform the division of x by G, then the leading monomial of the elements of G k[u] c can not divide any monomial of x k[u] and thus, they do not take part in the division. Hence, dividing x k[u] by G is exactly the same thing than dividing x by G k[u]. It then came naturally that G k[u] generates I k[u] since G generates I. In the same way, if we take S(f, g) with f, g G k[u], then S(f, g) k[u], and the division of S(f, g) by G k[u] is the same than the division by G. Hence the remainder is 0, and we have directly that G k[u] is a Gröbner basis of I k[u]. Remark. If U = {k,..., n} for some k {1,..., n}, then we can use the lexicographical order. And if U = {1,..., k}, we can also use the lexigraphical order, with the order in the variables taken backward. For the algorithms below, we will only need those two cases. 3.2 Intersection of Ideals With the ability of computing elimination ideals, we now can look at how intersection of ideals can be computed A formula The idea of the computation we give lies in the following propostion : Proposition 10. Let I and J be two ideals of k[x 1,..., X n ]. Let L = t I + (1 t) J in k[x 1,..., X n, t] (thus, we add a new variable, t), with the products formed by multiplying generators by t or 1 t, respectively. Then I J = L k[x 1,..., X n ]. 17

18 Proof. If x I J, then x = tx + (1 t)x with x I and x J, so x L, and by definition x k[x 1,..., X n ], thus x L k[x 1,..., X n ]. Conversely, if x L k[x 1,..., X n ], then x L, thus x = t i + (1 t) j for some i I, j J. Yet, x k[x 1,..., X n ], so we can evaluate x = t i + (1 t) j in t = 0 and t = 1 to find x = i = j and x I J. Thus the result is proven. Remark. We may notice that L k[x 1,..., X n ] is an elimination ideal. If theoritically, the formula seems easy to implemant, we still have to add a new variable, and not all computer algebra system handle that easily... In what follows, we will explain how it can be done in Magma How to add a new variable, coercion in Magma Magma can easily coerce polynomial from a polynomial ring to another, as soon as those two polynomial rings have the same rank... Hence, no automatic coercion is available between, say k[x 1,..., X n ] and k[y 1,..., Y n, t], or even with the use of so-called "global polynomial ring" of different rank... That is why, to implement successfully the intersection of ideals, we have to implement our own coercion functions. For that, hopefully, we can use Magma s homomorphisms : function expand(r,l) n:=rank(r); K:=BaseRing(R); R2:=PolynomialRing(K,n+l); Z:=[R2.i : i in [1..n]]; f:=hom<r->r2 Z>; return f,r2; function proj(r,l) n:=rank(r); K:=BaseRing(R); R2:=PolynomialRing(K,l); Z:=[R2.i : i in [1..l]] cat [0 : i in [(l+1)..n]]; f:=hom<r->r2 Z>; return f,r2; The first function, given a polynomial ring R = k[x 1,..., X n ], and a number l of variables to add, returns the canonical injection : f : k[x 1,..., X n ] k[y 1,..., Y n+l ], P P, and a polynomial ring R2 = k[y 1,..., Y n+l ]. 18

19 The second function, given a polynomial ring R = k[x 1,..., X n ], and a number l n of variables as a goal, returns the canonical surjection : fk[x 1,..., X n ] k[y 1,..., Y l ], X i Y i if i {1,..., l} and X i 0 elsewhere, and a polynomial ring R2 = k[y 1,..., Y l ]. Finally, if we have a polynomial ring R of rank n, if t N, expand will give us R 1 of rank n + t and f : R R 1, canonical injection. Then, proj(r 1,n) will give us R 2 of rank n and g : R 1 R 2 canonical surjection. Since R 2 is of rank n, we can use automatic coerction from R 2 to R. It will be performed by the command R!x which will coerce x in R, if possible. Hence, with f, g, and automatic coercion, we can indeed add t variables to R = k[x 1,..., X n ] when working with R 1 = k[y 1,..., Y n+t ] An Algorithm in Magma So, with those two functions, we can now write an algorithm in Magma to compute the intersection of two ideals : function intersection(f1,f2,r) n:=rank(r); n1:=#f1; n2:=#f2; f,r2:=expand(r,1); F3:=[f(F1[i])*R2.(n+1) : i in [1..n1]] cat [f(f2[i])*(1-r2.(n+1)) : i in [1..n2]]; G:=grobner(F3,R2,lexord); G2:=eliminationset(G,{n+1},R2); g,r3:=proj(r2,n); s:=#g2; G3:=[R!(g(G2[i])) : i in [1..s]]; return G3; 3.3 Ideal Quotient and Saturation Ideal Quotient With the previous algorithm to compute intersection of ideals, and with the two properties given below, we can easily deduce an algorithm to compute quotients of ideals. Proposition 11. Let I, J be two ideals of k[x 1,..., X n ], and let us write J = f 1,..., f k, and let f k[x 1,..., X n ]. Then we have : 19

20 I : J = k i=1 I : f i I : f = (I (f))f 1 The I : f = (I (f))f 1 means the ideal of the elements of I (f), divided by f. It is then rather straightforward to prove the proposition by just recalling the definitions... The algorithm follows directly the proposition Saturation Another interesting operation on ideal is that of saturation. Here is the definition of the saturation ideal of I with respect to f : Definition 21. I : f = i N I : f i is the saturation ideal of I with respect to f. Since the I : f i defines an ascending ideal chain, and with the fact that k[x 1,..., X n ] is noetherian, we can find k N such that I : f = I : f k The proposition below allows us to find a faster algorithm to compute saturation ideals than finding such a k. Proposition 12. Let I be an ideal of k[x 1,..., X n ], and let f k[x 1,..., X n ]. We define J =< I, tf 1 > an ideal of k[x 1,..., X n, t]. Then I : f = J k[x 1,..., X n ]. Proof. Let g J k[x 1,..., X n ]. Then g = q 1 i + q 2 (1 tf), with q 1, q 2 k[x 1,..., X n, t]. In k(x, Y ), we can substitute Y by 1 and then multiply the f equation by f d where d = deg t q 1 to obtain f d g = qi, where q k[x 1,..., X n ], thus g I : f. Conversely, let g I : f. Then there exists d N such that gf d I. By definition of J, 1 = tf + j with j J. Thus, 1 = (tf) d + j with j J. Thus, g = gf d t d + gj J, and we have proved the result. Again, the algorithm is just the application of the formula. 3.4 The Dimension of an Ideal In this section, we will explain how the dimension of an ideal can be computed. First, we will need some definitions Definitions and first properties Different definitions can be given to the dimension of an ideal. We will see that they somehow all corresponds. Definition 22. The Krull dimension of a ring is the supremum of the length n of strictly ascending chain I 0 I 1 I n of prime ideals of the ring. 20

21 The Krull dimension of an ideal I k[x 1,..., X n ] is the Krull dimension of the quotient ring k[x 1,..., X n ] I. If X is a topological space, we define the dimension of X as the supremum of the integers n such that there exists a strictly increasing chain Z 0 Z n of irreducible closed subset of X. We define the dimension of an affine variety as its dimension as a topological space. Proposition 13. If Y is an affine variety, then its dimension is equal to the dimension of I(Y ). Proof. The correspondance between prime ideals containing I(Y ) and closed irreducible subsets of Y shows us the correspondance between chains of prime ideals of k[x 1,..., X n ] I and chains of closed irreducible subsets of Y, and hence, the equality of the dimensions. Definition 23. If k L are two fields, and if S L, then S is called algebraically independant over k if for all {a 1,..., a n } S (no two the same), if P k[x 1,..., X n ] is such that P (a 1,..., a n ) = 0, then P = 0. The transcendence degree of L over k is the cardinal of the largest algebraically independant subset of L over k. The transcendence degree of L a k-algebra which is an integral domain is the transcendance degre of its fraction field. L is called a purely transcendental extension of k if there exists an algebraically independant over k S L such that L = k(s) We will need to admit the following theorem (related to Noether s Normalisation lemma). Its proof can be found in any good book about algebraic geometry. Theorem 8. If I is a prime ideal of k[x 1,..., X n ], then the dimension of I is equal to the transcendence degree of k[x 1,..., X n ] I over k. We shall also admit the link between the dimension of an ideal and that of its monomial ideal. Definition 24. A monomial ordering is called graded if deg(f) > deg(g) implies that f > g, with deg the total degree. Proposition 14. If I is an ideal of k[x 1,..., X n ], and if we take a graded monomial ordering, then dimi = dimlt (I). 21

22 3.4.2 Heuristics If k is an algebraically closed field, if we take G a Gröbner basis of an ideal I, with respect to a graded monomial ordering, then we can write I = g 1,..., g s, and LT (I) = X α 1, X α 1,n n,..., X α s, Xn αs,n with LM(gj ) = X α j, X α j,n n. Thus, with basic properties of operations on varieties, V (LT (I)) = i {1,...,s} = = V ( X α ) i, X α i,n n i {1,...,s} j {1,...,n} J {1,...,n} s i {1,...,s},α i,ji 0 V (X α i,j j ) = i {1,...,s} j {1,...,n},α i,j 0 V (X j ) = M L V (M), where L P ({X 1,..., X n }), using the distributivity of with. We shall admit the following proposition : V (X j ) Proposition 15. If V and W are affine varieties, then dim (V W ) = max (dimv, dimw ). Besides that, if M {X 1,..., X n }, then M is prime (k [{X 1,..., X n } \ M] being an integral domain) and tr deg k[x 1,..., X n ] M = n M. Hence, dimv (M) = n M. Thus, if V (LT (I)) = M L V (M), then dimi = max M L {n M}. We can notice that V (M) V (LT (I)) if and only if every generator of LT (I) involves at least one variable X i lying in M. Thus all the M can be combinatorialy computed, and thus the dimension. We can also notice that M = {X 1,..., X n }\M is such that V (M) V (LT (I)) if and only if any LM(g j ) involves at least one variable not in M, and then the dimension of I would be the maximum of the such M. This can be rephrase as M = {X 1,..., X n }\M is such that V (M) V (LT (I)) if and only if for all j, LM(g j ) / k[m ] An algorithm This is how the algorithm we give for the computation of the dimension of an ideal works : it reduces to the monomial ideal and computes all the possible M recursively starting with, and finally it gives the largest one and its cardinal. Those algorithms are purely combinatorial, and we will only refer to Becker and Weispfenning [6] (pages 449 to 451) for a proof. 22

23 Algorithm 3 Computation of the dimension of the ideal generated by F in R. G:=grobner(F,R,grevlexord); s:= G; for i in [1..s] do l:=normemultidegree(g[i],r,grevlexord); if l eq 0 then return -1; break i; else if l < 0 then Remove( G,i); end if end for return dim(g,r,grevlexord); Algorithm 4 dim function M:=dimrec(LTens(G,R,ord),1,{},{},R); d,u:=maxcardens(m); return M,d,U; 3.5 The Radical of an Ideal In this section, we will present how we can compute the radical of an ideal in any characteristic. We will first treat the case of the zero characteristic, and then that of the positive characteristic. Since those questions are not entirely easy, we will mostly only give few explanations about the algorithms, leaving the reader to look at the references for more information Zero-dimensional Radical in Zero Characteristic The algorithm we present for computing radical in zero characteristic relies first on the computation of radical of zero-dimensional ideals, which is simpler, and enough to deduce the computation for higher-dimensional ideals. The main idea for the computation comes from the following proposition : Proposition 16. Let I k[x 1,..., x n ] be an ideal (n 1). If I k[x i ] contains a separable polynomial for each i {1,..., }, then I = I. For a proof, we refer to Becker and Weispfenning [6] (lemmea 8.13). Then, the following algorithm, due to Faugère, allows us to compute univariate polynomials in a zero-dimensional ideal (here for the variable i) : 23

24 Algorithm 5 function dimrec(s,k,u,m,r) M2:=M; n:=rank(r); for i in [k..n] do if S k [{X 1,..., X n } \ (U {X i })] then M2:=dimrec(S,i+1,U join i,m2,r); end if end for if No element of M2 already contains U then M2:=M2 join U; end if return M2; Algorithm 6 function univariate(f,r,i) G:=grobner(F,R,grevlexord); n:=rank(r); d:=0; t,l d :=division(r.i d,g,r,grevlexord); {L is the remainder of the division} while L 0,..., L d is linearly independant do d:=d+1; t,l d :=division(r.i d,g,r,grevlexord); end while return d j=0 α j x j i where d j=0 α j L i = 0; 24

25 It is easy to see that if d j=0 α j L i = 0, then d j=0 α j x j i I, with the "linearity" of the remainder of the division by G. With the proposition and the algorithm, we can deduce an algorithm to compute the radical of a zero-dimensional ideal in caracteristic zero. Algorithm 7 function zerodimradical(f,r) n:=rank(r); for i in {1,..., n} do f i :=univariate(f,r,i); f i GCD(f i,f i ); {with the derivative with respect to x i} g i := end for return F cat [g 1,..., g n ] In zero characteristic, any of the g i is separable, and is of course in I, thus with the proposition, it is easy to see that I + g 1,..., g n = I, and it corresponds to the F cat [g 1,..., g n ] returned Higher-dimensional Radical in Zero Characteristic Before we can present the algorithm for any dimension radical, we shall present two proposition to explain how the algorithm works. Proposition 17. Let L = k (x r+1,..., x n ) be a rational function field and J be an ideal of L [x 1,..., x r ]. If G is a Gröbner basis of J with respect to any monomial ordering, such that G k [x 1,..., x n ], which can easily be achieved for any Gröbner basis by multiplying by the LCM of the denominators of the coefficients. Let f := LCM {LC(g), g G}, with the LCM taken in k [x r+1,..., x n ]. Let I be the ideal in k [x 1,..., x n ] generated by G, then J k [x 1,..., x n ] = I : f. Proposition 18. Let I k [x 1,..., x n ] be an ideal. Let G be a Gröbner basis with respect to the lexicographical ordering. Let f = LCM {LC(g), g G} where g is considering g as an element of k (x r+1,..., x n ) [x 1,..., x r ], and taking the leading coefficient with respect to the lexicographical ordering. Then the contraction ideal of J = Ik (x r+1,..., x n ) [x 1,..., x r ] is J k [x 1,..., x n ] = I : f. Furthermore, J = Ik (x r+1,..., x n ) [x 1,..., x r ] is zero-dimensional, with a k such that I : f = I : f k, then I = (I + (f k )) (I : f ). For a proof, we refer again to Becker and Weispfenning [6] (lemma 8.91, 8.94 and 8.95). With those two propositions, we can provide an algorithm to compute the radical ideal of an ideal in zero characteristic : 25

26 Algorithm 8 function radical0(f,r) n:=rank(r); M,d,U:=dimension(F,R); if d = -1 then return F; { F = k [x 1,..., x n ]} else if d=0 then return zerodimradical(f,r); else Renumber the variables such that M = {1,..., r} L := k (x r+1,..., x n ); with the previous proposition, find f k [x r+1,..., x n ] such that I = (I + (f k )) (IL [x 1,..., x r ] k [x 1,..., x n ] for some k J:=zerodimradical(IL [x 1,..., x r ],L [x 1,..., x r ]); with the previous proposition, compute J c = J k [x 1,..., x n ] I2:=radical0(I cat[f],r); return I2 J c ; end if end if Radical in positive characteristic Thanks to Matsumoto [8], there is a simpler algorithm for computing the radical of a polynomial ideal over a perfect positive-characteristic field. Yet, we will only refer to the article for proofs and explanations. 26

27 Algorithm 9 function radicalp(f,q,r), where q is a power of p, the characteristic of the base ring of R n:=rank(r); k:=basering(r); B:=grobner(F,R,grevlexord); trouve:=false; repeat B 0 := {φ(b i ), i {1,..., B}}; {where φ : α a α x α α a 1/q α x α which is well defined (perfect field)} B :=grobner(b 0 cat {Y 1 X1, q..., Y n Xn},k q [X 1,..., X n, Y 1,..., Y n ],lexord); B 0 :=elimination(b,n+1, k [X 1,..., X n, Y 1,..., Y n ]); {B k[y 1,..., Y n ]} B :=B 0 with substitution of the Y i by the X i ; if B = B then trouve:=true else B:=B ; end if until trouve 3.6 Computation Results and Examples Here, we will discuss a few exemple of use of our implementation of the algorithms previously given Basic operations First, we will see that X 2 Y XY 2 = X 2 Y 2 : > R:=PolynomialRing(RationalField(),2); > intersection([x^2*y],[x*y^2],r); [ X^2*Y^2 ] Then, we can compute XY (Z + 2) 2, (X + 3)Y 2 : XY Z, X(Y 1) 2 Z : > R:=PolynomialRing(RationalField(),3); > colon([x*y*(z+2)^2,(x+3)*y^2],[x*y*z,x*(y-1)^2*z],r); [ X*Y*Z^3 + 8*X*Y*Z^2 + 20*X*Y*Z + 16*X*Y, Y*Z^3 + 4*Y*Z^2 + 4*Y*Z, 27

28 -4*X*Y*Z^2-16*X*Y*Z - 16*X*Y, Y*Z^2 + 4*Y*Z + 4*Y, 12*X*Y^2*Z + 16*X*Y^2 + 36*Y^2*Z + 48*Y^2, 4/3*X*Y^2 + 4*Y^2 ] We can also compute X 2 + Z, Y 2 + Z : (X Y ) : > R:=PolynomialRing(RationalField(),3); > saturation([x^2+z,y^2+z],x-y,r); [ X^2 + Z, Y^2 + Z, X + Y ] About Dimension Computation Now we will take a quick look at what our implementation for the computation of the dimension of an ideal is able to do : > R:=PolynomialRing(RationalField(),3); > F:=[Y^2*Z^3,X^5*Z^4,X^2*Y*Z^3]; > G:=grobner(F,R,grevlexord); > dimension(g,r); { { 3 }, { 1, 2 } } 2 { 1, 2 } The answers given are, in that order : The list of all maximal independant subset of {X 1,..., X n } ; The dimension of the ideal ; A maximal independant subset of maximal cardinal. Yet, if the polynomials generating the ideal grow in degree (say, total degree), we will see how our implementation is limited : 28

29 > dimension(grobner([(x-1)^2*y*z,x*(y+2)^2*(z+3), X*(Y+5)*(Z+2)],R,grevlexord),R); { { 2 }, { 3 } } 1 { 3 } > dimension(grobner([(x-1)^2*y*z,x*(y+2)^2*(z+3), (X+3)*(Y+5)*(Z+2)],R,grevlexord),R); { {} } 0 {} > dimension(grobner([(x-1)^2*y^3*z^4,x*(y+2)^2*(z+3)^3, (X+3)^2*(Y+5)^3*(Z+2)],R,grevlexord),R); And the last one would not give any answer in one hour About Radical Computation Here is an exemple of the computation of the radical of a radical ideal. We see that indeed, even if we get a more complicated set of generators as an answer (due to Gröbner basis computation), we indeed get the same ideal : > R:=PolynomialRing(RationalField(),3); > I:=ideal; > F0:=[x*y*z,(x+1)*(y+1)*(z+1),(x+2)*(y+2)*(z+2)]; > radical(f0,r); [ x*y*z, x*y*z + x*y + x*z + x + y*z + y + z + 1, x*y*z + 2*x*y + 2*x*z + 4*x + 2*y*z + 4*y + 4*z + 8, -x*y - x*z - x - y*z - y - z - 1, -2*x - 2*y - 2*z - 6, z^3 + 3*z^2 + 2*z, -y^2*z - y*z^2-3*y*z, -y^2 - y*z - 3*y - z^2-3*z - 2, 1/2*x^3 + 3/2*x^2 + x, 1/2*y^3 + 3/2*y^2 + y, 1/2*z^3 + 3/2*z^2 + z 29

30 ] > A:=radical(F0,R); > equal(a,f0,r); true > IsRadical(I); true We can consider the computation of the radical of more complicated ideals : > R:=PolynomialRing(RationalField(),2); > radical([x^2*y^2],r); [ x^2*y^2, x*y^2, x^2*y, x*y ] > radical([x^2*y^2,(x+1)^3*(y-1)],r); [ x^2*y^2, x^3*y - x^3 + 3*x^2*y - 3*x^2 + 3*x*y - 3*x + y - 1, x^3 + 3*x^2-3*x*y^2 + 3*x - y^2 + 1, 3*x*y^4-3*x*y^2 + y^4 - y^2, 3*x*y^3-3*x*y^2 + y^3 - y^2, 1/9*y^4-1/9*y^2, 1/9*y^3-1/9*y^2, x^2 + x, -y^2 + y ] > I:=ideal<R x^2*y^2,(x+1)^3*(y-1)>; > Radical(I); Ideal of Polynomial ring of rank 2 over Rational Field Lexicographical Order Variables: x, y Dimension 0, Radical Groebner basis: [ x - y + 1, y^2 - y ] > J:=radical([x^2*y^2,(x+1)^3*(y-1)],R); 30

31 > equal(j,[x-y+1,y^2-y],r); true > I:=ideal<R x^2*y^2,(x+1)^3*y>; > Dimension(I); 1 [ 1 ] > radical([(x*y)^2,(x+1)^3*y],r); [ x^2*y^2, 3*x*y^2 + y^2, x^3*y + 3*x^2*y + 3*x*y + y, -1/9*y^2, -3*x^2*y - 3*x*y - y, 9*x*y + 5*y, y ] > J:=radical([(x*y)^2,(x+1)^3*y],R); > Radical(I); Ideal of Polynomial ring of rank 2 over Rational Field Lexicographical Order Variables: x, y Radical Groebner basis: [ y ] > equal(j,[y],r); true Finally, in positive characteristic, we can see that Matsumoto s algorithm seems very efficient : > T<u,x,y,z>:=PolynomialRing(GF(7),4); > F:=[z^7-x*y*u^5,y^4-x^3*u]; > radical(f,t); [ 6*u*x + y*z, u*y^2 + 6*x*z^2, 6*u^2*y + z^3, x^2*z + 6*y^3 ] > I:=ideal<T z^7-x*y*u^5,y^4-x^3*u>; 31

32 > I; Ideal of Polynomial ring of rank 4 over GF(7) Lexicographical Order Variables: u, x, y, z Basis: [ 6*u^5*x*y + z^7, 6*u*x^3 + y^4 ] > Radical(I); ^C [Interrupted] Indeed, Matsumoto s algorithm, as I have implemented it on Magma, can compute radical of ideals in positive characteristic that even Magma can not compute! Well at least, its version V don t seem to be able to compute any positivecharacteristic radical... 4 Computational Invariant Theory 4.1 Symmetric Polynomials Invariant Theory is a very anciant topic, beginning with symmetric polynomials : everyone has heard of the fundamental theorem of symmetric polynomials (while not everyone has seen a proof of this theorem...), but we will see that the theory is far much deeper, going to recent developpements. In this section, we will specially consider Computational Invariant Theory, and some of his connection with Gröbner basis. We will indeed begin with symmetric polynomials, and give a proof the fundamental theorem of symmetric polynomials Some definitions Definition 25. We will note, if n N, A M n (k), f k [X 1,..., X n ], f (A.X) for f ( n k=1 a 1,k X k,..., n k=1 a n,k X k ), f (A.X) k [X 1,..., X n ]. Definition 26. We define the elementary symmetric functions σ 1,..., σ n of k [x 1,..., x n ] by : σ 1 = x x n,..., σ r = i 1 < <i r x i1... x ir,..., σ n = x 1... x n. We will need a little lemma, which will soon reveal useful in some of the proofs below : 32

33 Lemma 12. Let σ (n) i be the ith elementary symmetric function in variables x 1,..., x n, with n N and i {1,..., n}. We set σ (n) 0 = 1 (for all n N ) and σ (n) i = 0 if i < 0 or i > n (i Z). Then n N, i Z, σ (n) i = σ (n 1) i + x n σ (n 1) i 1. Proof. Let n N and r Z. If r 0 or r > n, then we indeed have σ r (n) If r = n, then σ n (n) equality still holds. = x 1... x n, σ (n 1) n = σ (n 1) r = 0 and x n σ (n 1) n 1 + x n σ (n 1) r 1. = x 1... x n 1 x n, so the If r = 1, then σ (n) 1 = x x n, σ (n 1) 1 = x x n 1 and x n σ (n 1) 0 = x n and again, there is no problem. Finally, if r < n then : So, and the result is proven. σ (n) r = i 1 < <i r x i1... x ir. σ (n 1) r = x n σ (n 1) k 1 = σ (n 1) r i 1 < <i r<n i 1 < <i r 1 <n + x n σ (n 1) r 1 = x i1... x ir. x i1... x ir 1 x n. i 1 < <i r x i1... x ir = σ (n) r We set ( 1) 0 = 1. We are now able to prove this first very famous result : Proposition 19. It is well-known that (X x 1 )... (X x n ) = n i=0 ( 1) i σ i X n i, n N (with here σ i = σ (n) i ). Proof. If n = 1, with ( 1) 0 = 1, ( 1) 0 σ (1) 0 X + ( 1) 1 σ (1) 1 X 0 = X x 1 and the result is proven. We assume that the result holds for n 1 (n > 1). Then with the previous lemma : 33